Geometry Craft

[vampirewitchreine]

(This was in an old section: Conditions for Special Parallelograms)

Two common crafts in Japan are paper-folding (origami) and paper-cutting (kirigami). Since both crafts usually require a square sheet of paper, it is helpful to know how to cut a nonsquare rectangular sheet of paper into a square.
Step 1: Fold the paper so that a short edge lines up with a long edge.
Step 2: Then cut along the edge of the two-layer triangle.
Why is the resulting rectangle a square? Which theorem does this process illustrate?

My theorems:
Ways to prove that a Parallelogram is a Rectangle

• If the diagonals of a parallelogram are congruent, then it is a rectangle
  
• If one angle of a parallelogram is a right angle, then it is a rectangle
  

Ways to prove that a Parallelogram is a Rhombus

• If the diagonals of a parallelogram are perpendicular, then it is a rhombus
  
• If two consecutive sides of a parallelogram are congruent, then it is a rhombus
  

I did this a couple of ways.I did this with a piece of paper and a math program that I have downloaded. Perhaps I'm over thinking this or perhaps overlooking something totally obvious, but I'd greatly appreciate any help that you can give me here.)

The following is what I did on my program (The colors were only to help know what happened..... the dark blue color is the extra rectangle that gets cut off and the purple/pink is the square, with the 2 colors showing that it was folded into two congruent triangles to make it into a square):

 

[Mrspi]

(This was in an old section: Conditions for Special Parallelograms)

Two common crafts in Japan are paper-folding (origami) and paper-cutting (kirigami). Since both crafts usually require a square sheet of paper, it is helpful to know how to cut a nonsquare rectangular sheet of paper into a square.
Step 1: Fold the paper so that a short edge lines up with a long edge.
Step 2: Then cut along the edge of the two-layer triangle.
Why is the resulting rectangle a square? Which theorem does this process illustrate?

My theorems:
Ways to prove that a Parallelogram is a Rectangle

• If the diagonals of a parallelogram are congruent, then it is a rectangle
  
• If one angle of a parallelogram is a right angle, then it is a rectangle
  

Ways to prove that a Parallelogram is a Rhombus

• If the diagonals of a parallelogram are perpendicular, then it is a rhombus
  
• If two consecutive sides of a parallelogram are congruent, then it is a rhombus
  

I did this a couple of ways.I did this with a piece of paper and a math program that I have downloaded. Perhaps I'm over thinking this or perhaps overlooking something totally obvious, but I'd greatly appreciate any help that you can give me here.)

The following is what I did on my program (The colors were only to help know what happened..... the dark blue color is the extra rectangle that gets cut off and the purple/pink is the square, with the 2 colors showing that it was folded into two congruent triangles to make it into a square):
View attachment 1578

You are definitely overthinking this.

You STARTED with a piece of paper which was a rectangle, right? So you know that it has right angles already. If you follow the steps described, and cut off that dark blue rectangle, you will have four equal sides in your figure.

Four right angles....rectangle.
Four equal sides....rhombus.

If a figure is both a rectangle and a rhombus, then it is also a ......

 

[wjm11]

...how to cut a nonsquare rectangular sheet of paper into a square.
Step 1: Fold the paper so that a short edge lines up with a long edge.
Step 2: Then cut along the edge of the two-layer triangle.
Why is the resulting rectangle a square? Which theorem does this process illustrate?

Here is one way:

You have two triangles , one atop the other, so it is easy to prove congruency -- by side/angle/side, for example.

Next, demonstrate that they are isosceles right triangles. The acute angles must, therefore, be equal to 45 degrees each.

That should do it; you now have four equal sides and right angles in the corners.

Make sense?

 

[vampirewitchreine]

You are definitely overthinking this.

You STARTED with a piece of paper which was a rectangle, right? So you know that it has right angles already. If you follow the steps described, and cut off that dark blue rectangle, you will have four equal sides in your figure.

Four right angles....rectangle.
Four equal sides....rhombus.

If a figure is both a rectangle and a rhombus, then it is also a ......

Thank you (It think my biggest thing was that I was looking for ONE particular theorem, rather than using a combination of them.)


Given the information that you gave me, I have the following answer to my question:
Since one of the angles in this parallelogram is a right angle, the resulting figure is a rectangle. Due to two consecutive sides being congruent, the resulting figure is a rhombus. Finally, because the figure is both equilateral and equiangular the figure is a square.

Here is one way:

You have two triangles , one atop the other, so it is easy to prove congruency -- by side/angle/side, for example.

Next, demonstrate that they are isosceles right triangles. The acute angles must, therefore, be equal to 45 degrees each.

That should do it; you now have four equal sides and right angles in the corners.

Make sense?

It does. When I first read this, I had a little bit of a hard time following you (I'm not very awake right now). But I went back and re-read this a couple of times and then understood what you were saying. Thank you for taking the time to show me that way as well, but Mrspi helped me out by using the theorems that I posted from the book (which I was suppose to use to prove with). If I have a similar problem in the future that doesn't require the theorems, I will definitely keep this way in mind.

Edit: I just came across an instance in which this might be useful if you wouldn't mind helping me with it.


13. Quilting Many quilts feature squares that are made up of triangles. Explain why placing four congruent isosceles right triangles as shown always result in a square. (I created my own image rather than fight my webcam, scanner and Google images. The light colored box is suppose to highlight the square of the quilt as shown in my book)

(Please keep in mind that this falls well before special right triangles, in the same section as the previous question in this thread)